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arXiv · 1912.08945

Tunnel number and bridge number of composite genus 2 spatial graphs

Scott A. Taylor·Maggy Tomova
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Abstract

Connected sum and trivalent vertex sum are natural operations on genus 2 spatial graphs and, as with knots, tunnel number behaves in interesting ways under these operations. We prove sharp Scharlemann-Schultens type bounds for the tunnel number of a composite genus 2 spatial graph. For the tunnel number of a composite Brunnian $θ$-curve, our result implies that the tunnel number is at least the number of summands, as in the knot case. We also prove a version of a theorem of Morimoto for knots: the tunnel number of a composite m-small genus 2 spatial graph is at least the sum of the tunnel numbers of the factors. We also study lower bounds for the bridge number of composite genus 2 graphs. In particular, our results imply that for a Brunnian composite $θ$-curve having $m$ factors in its prime factorization, the bridge number is at least $m+3/2$.

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Scott A. Taylor, Maggy Tomova. 2021-06-28. Tunnel number and bridge number of composite genus 2 spatial graphs. https://doi.org/10.2140/pjm.2021.314.451

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